Question: $ C = \left[\begin{array}{rr}-1 & 5 \\ 0 & -2\end{array}\right]$ $ F = \left[\begin{array}{rrr}5 & -1 & 3 \\ 3 & 2 & 4\end{array}\right]$ What is $ C F$ ?
Solution: Because $ C$ has dimensions $(2\times2)$ and $ F$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ C F = \left[\begin{array}{rr}{-1} & {5} \\ {0} & {-2}\end{array}\right] \left[\begin{array}{rrr}{5} & \color{#DF0030}{-1} & \color{#9D38BD}{3} \\ {3} & \color{#DF0030}{2} & \color{#9D38BD}{4}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{-1}\cdot{5}+{5}\cdot{3} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{-1}\cdot{5}+{5}\cdot{3} & ? & ? \\ {0}\cdot{5}+{-2}\cdot{3} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{-1}\cdot{5}+{5}\cdot{3} & {-1}\cdot\color{#DF0030}{-1}+{5}\cdot\color{#DF0030}{2} & ? \\ {0}\cdot{5}+{-2}\cdot{3} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{-1}\cdot{5}+{5}\cdot{3} & {-1}\cdot\color{#DF0030}{-1}+{5}\cdot\color{#DF0030}{2} & {-1}\cdot\color{#9D38BD}{3}+{5}\cdot\color{#9D38BD}{4} \\ {0}\cdot{5}+{-2}\cdot{3} & {0}\cdot\color{#DF0030}{-1}+{-2}\cdot\color{#DF0030}{2} & {0}\cdot\color{#9D38BD}{3}+{-2}\cdot\color{#9D38BD}{4}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}10 & 11 & 17 \\ -6 & -4 & -8\end{array}\right] $